ŌÓÁŤŲŤÓŪŤūÓ‚ŗŪŤŚ ŠŤÁŪŚŮŗ

Identifying what Kelvin called the ‚Ä˜molecular tactics of a crystal‚Ä™ remained a
hesitant and erroneous process until x-rays provided the means to determine these
Atoms and molecules: begging the question 77

Figure 8.2. The four elements and the Universe in Plato‚Ä™s conception (from a drawing by
J Kepler).

arrangements reliably if not quite directly, early in the 20th century. Today we
can Ô¬Ānally see (or, more accurately, feel ) the individual atoms on the surface of a
crystal, using the scanning tunnelling or atomic force microscope.

8.2 Atoms and molecules: begging the question

Whether matter is discrete or continuous has been a subject of debate at least since
the time of the ancient Greek philosophers. The Ô¬Ārst detailed atomistic theory was
that of Plato (in the Timaeus) who described matter to be ‚Ä˜One single Whole, with
all its parts perfect‚Ä™. He associated the four ‚Ä˜elements‚Ä™‚Ä”earth, water, Ô¬Āre and
air‚Ä”with the form of four regular polyhedra‚Ä”cube, icosahedron, tetrahedron
and octahedron‚Ä”and with the dodecahedron he associated the Universe. Plato
attempted to match the properties of the elements with the shapes of the constitu-
tive atoms. For instance water, being Ô¬‚uid, was associated with the icosahedron,
which is the most spherical among the Ô¬Āve regular solids. This theory was able
to offer an ad hoc explanation of phase transitions (for instance the transition
solid-to-liquid-to-vapour, which means earth-to-water-to-air and corresponds to
cube-to-icosahedron-to-octahedron).
The atomistic theory of Plato was dismissed by Aristotle. He argued that if
the elements are made up of these particles then the copies of each regular poly-
hedra must Ô¬Āll the space around a point and this operation cannot be done with
the icosahedron and the octahedron (he thought erroneously that it was possible
to Ô¬Āll space with regular tetrahedra).
The architecture of the world of atoms
78

In other periods atoms have been reduced to mere points, or considered to
be hard or soft spheres, or to have more exotic shapes endowed with speciÔ¬Āc
properties inspired by chemistry. As Newton said, the invention of such ‚Ä˜hooked
atoms‚Ä™ by followers of Descartes often begged the question.
Christian Huygens in his Trait¬ī de la lumi` re (1690) suggested that the Ice-
e e
land spar (at that time very much studied for its birefringence) may be composed
of an array of slightly Ô¬‚attened spheroids (see Ô¬Āgure 8.3(c)). In this way he ex-
plained in one stroke the birefringence and the cleavage properties of these crys-
tals.
Ha¬® y‚Ä™s celebrated constructions begged the question, in as much as he ex-
u
plained the external form of crystals as being due to the packing of small compo-
nents which were identical to the crystal itself. Ha¬® y‚Ä™s 1784 Essai d‚Ä™une theorie
u
sur la structure des crystaux was nevertheless the most perspicacious of early at-
tempts to make sense of crystals, in that he recognized that their angles are not
arbitrary but follow certain rules, still used today.

8.3 Atoms as points
The Newtonian vision was taken to an extreme by Roger Joseph Boscovitch, early
in the 18th century. He postulated that ‚Ä˜matter is composed of perfectly indivisi-
ble, non-extended, discrete points‚Ä™ which interacted with one another.
Boscovitch published his Theory of Natural Philosophy on 1758, when he
was a professor at the Collegium Romanum. He has been described as a philoso-
pher, astronomer, historian, engineer, architect, diplomat and man of the world.
Given all this, the book is a disappointingly dry exposition in which he attempts
to deduce much of physics from a ‚Ä˜single law of forces‚Ä™. This means a mutual
force between each pair of points. Boscovitch struggled to describe a possible
form for this, drawing illustrations which resemble modern graphs of interatomic
interactions.
Half a century later, French mathematicians such as Navier and Cauchy de-
veloped powerful theories of crystal elasticity based on this idea but independent
of any particular form for the interaction. Their elegant analysis of the effect of
crystal symmetry on properties has provided one of the enduring strands of phys-
ical mathematics. However, they were little concerned with the origins of crystal
structure itself.
Late in the 19th century, the Irish physicist Joseph Larmor reduced the role
of matter still further, to a mere mathematical singularity in the ether. His view
was enshrined in the book Aether and Matter, to which contemporaries jokingly
referred as ‚Ä˜Aether and No Matter‚Ä™. It was published in 1900, a date after which
Larmor declared that all progress in physical science had ceased. He was probably
not serious‚Ä”it is, in fact, the date of the inception of quantum theory, which
Ô¬Ānally told us what atoms are really like.
Today‚Ä™s quantum mechanical picture of a nucleus surrounded by a cloud
Atoms as points 79

of electrons is a subtle one: it taxes the resources of the largest computers to
predict what happens when these clouds come into contact. The day has not yet
come when older, rough-and-ready descriptions are completely obsolete. It is still
useful for some purposes to picture atoms as hard balls with relatively weak forces
of attraction pulling them together. In particular the structures of many metals can
be understood in this way.
The architecture of the world of atoms
80

Figure 8.4. All the snow crystals have a common hexagonal pattern and most of them
show a hexagonal shape.

8.4 Playing hardball

Among the manifold older ideas about atoms, the elementary notion of a hard
sphere has endured as a useful one, even today.
Spherical ‚Ä˜atoms‚Ä™ were adopted by Kepler to explain the hexagonal shape of
snowÔ¬‚akes (chapter 3). Kepler assumed that they were composed of tiny spheres
arranged in the plane in the triangular packing. He did not consider these spheres
to be atoms in the modern sense but more as the smallest particle of frozen water.
Of the later re-inventions of this type of theory, one of the most inÔ¬‚uential
was that of William Barlow, writing in Nature in 1883.
Barlow is a prime example of the self-made scientist. His paper ‚Ä˜On the
probable nature of the internal symmetry of crystals‚Ä™ is remarkable for its total
lack of any reference to previous work. He happily ignored centuries of specula-
tions, particularly those of his compatriot William Wollaston (see Ô¬Āgure 8.3) early
in the same century. His inÔ¬‚uence, particularly on and through Lord Kelvin, may
be attributed in part to his clear writing style, his choice of the simplest cases and
his use of attractive illustrations (Ô¬Āgure 8.5).
Barlow‚Ä™s method was to look for dense packings of spheres, with no attempt
at proof that they were the densest. This he left for mathematicians to consider
later. He mentioned the possibility that soft spheres might make more realistic
atoms but did little with this.
His down-to-earth commonsense approach might be regarded as a reaction
against the reÔ¬Āned mathematics of the French school, which many British natural
philosophers, such as Tait, had found rather indigestible. John Ruskin, venturing
into science in a manner which was then fashionable, had said in his Ethics of the
Dust (1865), which consisted of ‚Ä˜ten lectures to little house wives on the elements
of crystallization‚Ä™, that the ‚Ä˜mathematical part of crystallography is quite beyond
girls‚Ä™ strength‚Ä™. One might suppose that it would be beyond Barlow‚Ä™s strength as
Playing hardball 81

Figure 8.5. Some illustrations from Barlow‚Ä™s papers.

well, but in fact he took it up avidly, and absorbed the full mathematical theory in
later years. Indeed he published in that area (albeit with some further disregard
for the precedents).
By 1897 he was ready to expound a more mature version of the theory in
a more erudite style. He described many possible stackings for hard spheres, of
equal or unequal sizes.
Barlow‚Ä™s intuitive attack scored a number of notable hits, particularly in pre-
The architecture of the world of atoms
82

Figure 8.6. One of the earliest x-ray photographs (1896).

dicting the structures of the alkali halides. His place in the history of science was
then assured by a contemporary and apparently unrelated discovery.
On 8 November 1895 a professor of physics in W¬® rzburg realized that a new
u
type of ray was emanating from his discharge tube. The R¬® ntgen ray, named after
o
its discoverer but eventually to be called the x-ray, was an immediate popular
sensation, much to R¬® ntgen‚Ä™s distaste. The potential for medical science and
o
the challenge to the modesty of Victorian ladies was clear‚Ä”the implications for
physics were not. One of the most extraordinary of these, which took two decades
to emerge, was the determination of crystal structures using x-rays, vindicating
much of the guesswork of Barlow and others.
It should not be thought that all crystal structures are dense packings of balls.
In the structure of diamond, each atom has only four atoms as neighbours. And
this does not even qualify as a stable loose packing. This structure too was pre-
saged before it was observed‚Ä”this time by Walter Nernst (1864‚Ä“1941), better
known for his Heat Theorem (the Third Law of Thermodynamics).

8.5 Modern crystallography
In a crystal the structure repeats a local conÔ¬Āguration of atoms as in a three-
dimensional wall paper. There are 14 ways to construct such periodic structures
in three dimensions, the Bravais lattices, but 230 different types of internal sym-
metry. This ‚Ä˜crushingly high number of 230 possible orderings‚Ä™, as Voigt called it,
was both challenging and depressing to the theorist, until x-ray diffraction offered
Crystalline packings 83

the means to use the theory in every detail.
When x-rays (electromagnetic radiation with a typical wavelength between
ňö
0.1 and 10 Angstroms, i.e. between 0.000 000 01 and 0.000 001 mm) are incident
on a crystal, they are diffracted and form a pattern with sharp spots of high inten-
sity corresponding to speciÔ¬Āc angular directions. On 8 June 1912 at the Bavarian
Science Academy of Munich, a study entitled ‚Ä˜Interference effects with R¬® ntgen o
2
rays‚Ä™ was presented . In this work Max von Laue developed a theory for the
diffraction of x-rays from a periodic packing of atoms, associating the spots of
intensity in the diffraction pattern with the regularity of the positions of the atoms
in the crystal structure. One year later Bragg reported the Ô¬Ārst determination of
crystal structures from x-ray diffraction for such systems as KCl, NaCl, KBr and
KI, conÔ¬Ārming Barlow‚Ä™s models 3 .

8.6 Crystalline packings

In many crystal structures one or several types of atom are in positions corre-
sponding to the centres of spheres in a sphere packing, as Barlow had supposed.
It has been noted by O‚Ä™Keeffe and Hyde 4 , two experts in crystal structures, that ‚Ä˜it
is hard to invent a simple symmetric sphere packing that does not occur in nature‚Ä™.
Of these, the most important family is that of packings with the maximal
¬ľ¬ľ
density of which the face-centred cubic (fcc) is a member. This
maximal density can be realized in inÔ¬Ānitely many ways, all of which are based
on the stacking of close-packed layers of spheres (Ô¬Āgure 8.8), as practised by the
greengrocer.
Two possibilities present themselves for the relative location of the next
layer, if it is to Ô¬Āt snugly into the Ô¬Ārst one. Each successive layer offers a similar
choice, and only by following a particular rule will the fcc structure described by
Kepler result, with a cubic symmetry, which is not at all apparent in this building
procedure. A different rule produces the hexagonal close-packed (hcp) structure,
also described by Barlow, the next in order of simplicity. These two structures
occur widely among the structures formed by the elements of the Periodic Table.
More complex members of the same family‚Ä”for example the double hexagonal
close-packed structure‚Ä”are found in alloys.
The original appeal of crystals lay in their external shapes, and these pro-
vided clues to their internal order. However, the precise shape of a crystal in
equilibrium cannot be deduced from this order alone. According to a principle
enunciated by Gibbs and Curie in 1875, the external shape of a crystal minimizes
¬ĺ
Friedrich W, Knipping P and Laue M 1912 Interferenz-Erscheinungen bei Roentgenstrahlen S. B.
Bayer. Akad. Wiss. pp 303‚Ä“22.
¬Ņ
Bragg W L 1913 The structure of some crystals as indicated by their diffaction Proc. Roy. Soc. A
89 248‚Ä“77.
O‚Ä™Keefe M and Hyde G B 1996 Crystal Structures (Washington, DC: Mineralogical Society of
America).
The architecture of the world of atoms
84

the total surface energy. This is made up of contributions from each facet but
different types of facet are more expensive in terms of energy.
Some of the observed shapes can be realized by a rule developed by Bravais:
the largest facets have the densest packing of atoms (which might be expected to
have the lowest surface energy).
If the packing fraction is decreased a little from its maximum value, allowing
the hard spheres some room to move, and they are given some kinetic energy, what
can be said about the competition between these structures? This is a very delicate
question for thermodynamics, and it has only been settled recently by extensive
computations. The winner (as the more stable structure) is fcc 5 .
Some natural elements which have an fcc or hcp crystalline structure are
given in table 8.1.

Figure 8.8. The hexagonal closed-packed (hcp) (a) and cubic closed-packed (fcc) (b)
structures. These lattices are generated by a sequence of layers of spheres in the triangular
packing conÔ¬Āguration (c). Suppose that the Ô¬Ārst layer of spheres has centres in position
A, the second layer can be placed in position B (or equivalently C), and for the third layer
we have two alternatives: (i) placing the centres of the spheres in position A generating
the sequence ABABAB. . . (which corresponds to the hcp lattice); (ii) placing the centres
in position C generating the sequence ABCABCABC. . . (which corresponds to the fcc
lattice).

8.7 Tetrahedral packing
Regular tetrahedra cannot pack together to Ô¬Āll space but irregular ones may do
so. Of special interest for crystal chemistry are packings in which neighbouring
atoms are on the vertices of such a system of closely packed tetrahedra. These
structures are called ‚Ä˜tetrahedrally packed‚Ä™.
A very important tetrahedrally-packed structure is the body-centred cubic
lattice (bcc). This is the crystalline structure of many chemical elements. The
bcc structure is the only tetrahedrally packed structure where all tetrahedra are
identical.
The architecture of the world of atoms
86

A special class of structures consists of those in which the packing is re-
stricted to conÔ¬Āgurations in which Ô¬Āve or six tetrahedra meet at an edge. These
are the crystal structures of some of the more important intermetallic phases. Such
structures are known as tetrahedrally close packed (tcp) and were described in the
1950s by Frank and Kasper 6 . These two eminent crystallographers were inspired
to produce their classiÔ¬Ācation of complex alloy structures by a visit to Toledo,
where the Moorish tiling patterns incorporate subtle mixtures of coordination
and symmetry, particularly Ô¬Āvefold features such as the regular pentagon. In the
Frank‚Ä“Kasper structures the packed spheres have various combinations of 12, 14,
15 and 16 neighbours, and the average is between 13.33. . . and 13.5.
Each tetrahedron of a tetrahedrally-packed structure has four other tetrahe-
dra sharing its faces, so the dual structure, derived by placing the vertices in the
centre of the tetrahedra (i.e. in the holes between spheres), forms a four-connected
network. Such a network can be regarded as the packing of polyhedra which are

8.8 Quasicrystals
We have seen in the previous paragraphs how, starting from such clues as the regu-
lar angular shapes of crystals, scientists constructed a theory‚Ä”crystallography‚Ä”
in which the intrinsic structure of crystals was described as a periodic assembly
of atoms, in an apparently complete system of ordered structures in the solid
state. But in November 1984 a revolution took place: Shechtman, Blech, Gratias
and Chan identiÔ¬Āed, in rapidly solidiÔ¬Āed AlMn alloys, an apparently new state
of condensed matter, ordered but not periodic. The scientiÔ¬Āc journal Physics To-
day headlined ‚Ä˜Puzzling Crystals Plunge Scientists into Uncertainty‚Ä™. Marjorie
Senechal, a mathematician who has been one of the protagonists of this revolu-
tion, described this climate of astonishment in her book Quasicrystals and Geom-
etry7 :
It was evident almost immediately after the November 1984 announce-
ment of the discovery of crystals with icosahedral symmetry that new
areas of research had been opened in mathematics as well as in solid
state science. For nearly 200 years it had been axiomatic that the in-
ternal structure of a crystal was periodic, like a three-dimensional wall-
paper pattern. Together with this axiom, generations of students had
learned its corollary: icosahedral symmetry is incompatible with peri-
odicity and is therefore impossible for crystals. Over the years, an ele-
gant and far-reaching mathematical theory had been developed to inter-
pret these ‚Ä˜facts‚Ä™. But suddenly‚Ä”in the words of the poet W B Yeats‚Ä”
all is changed, changed utterly.
What terrible beauty was born? These new solids, with diffraction patterns
which exhibit symmetries which are forbidden by the crystallographic restric-
tions, have been called quasicrystals. The internal structure of a quasicrystal is
an ordered packing of identical local conÔ¬Āgurations with non-periodic positions
in space.
With this perspective, it is a shock to realize that the seed of quasi-crystallin-
ity was already there in Kepler‚Ä™s work 8 . In his book Harmonices Mundi (1619)
Kepler described a repetitive structure with the ‚Ä˜forbidden‚Ä™ Ô¬Āvefold symmetry, but
with ‚Ä˜certain irregularities‚Ä™.
If you really wish to continue the pattern, certain irregularities must be
admitted, (. . . ) as it progresses this Ô¬Āve-cornered pattern continually
introduces something new. The structure is very elaborate and intricate.
Senechal M 1995 Quasicrystals and Geometry (Cambridge: Cambridge University Press).
Or even older instances in Islamic architecture: see Makovicky E 1992 Fivefold Symmetry ed I Har-
gittai (Singapore: World ScientiÔ¬Āc) p 67.
The architecture of the world of atoms
88

(a) (b)

Figure 8.9. Two non-periodic tilings: that proposed by Kepler in 1619 (a) and that pro-
posed by Penrose in 1974 (b). Note that Kepler‚Ä™s one is Ô¬Ānite whereas Penrose‚Ä™s can be
continued on the whole plane.

Hundreds of years later, in 1974, Roger Penrose produced a tiling that can
be considered the realization of the one described by Kepler (see Ô¬Āgure 8.9) 9.
The Penrose tiling covers the entire plane; it is non-periodic but repetitive.
Non-periodic means that if one takes two identical copies of the structure there
is only one position where these two structures superimpose perfectly. In other
words, sitting in a given position, the landscape around, up to an inÔ¬Ānite distance,
is unique and cannot be seen from any other point. Repetitive means that any local
part of the structure is repeated an inÔ¬Ānite number of times in the whole structure.
Exactly ten years after Penrose‚Ä™s work, this pattern was Ô¬Ārst observed in
nature.
What most surprised the researchers after the discovery of quasicrystals was
that these structures have diffraction patterns with well-deÔ¬Āned sharp peaks that
were previously considered to be the signature of periodicity. How can aperiodic
structures have sharp diffraction peaks? Let us just say that the condition to have
diffraction is associated with a strong form of repetitiveness, which is typical of
these quasicrystalline structures.
Mathematically, these quasiperiodic patterns can be constructed from a crys-
talline periodic structure in a high-dimensional hyperspace by cutting it with a
plane oriented with an irrational angle in respect to the crystalline axis. This con-

struction explains the existence of diffraction peaks but does not give any physical
understanding.
Why has nature decided to pack atoms in these non-periodic but repetitive
quasicrystalline conÔ¬Āgurations? This is a matter of debate, unlikely to be quickly
resolved10.
As a consequence of this revolution it was necessary for the scientiÔ¬Āc com-
munity to ask: what is to be considered to be a crystal?
The International Union of Crystallography established a commission on
Aperiodic Crystals that, in 1992, proposed for ‚Ä˜crystal‚Ä™ the following deÔ¬Ānition:

A crystal is any solid with an essentially discrete diffraction diagram.

According to this edict, the deÔ¬Ānition of a crystal which we have given earlier is
too restrictive but the new one will, alas, be a mystery to many.

8.9 Amorphous solids
Since ancient times, the distinction has been made between crystals and non-
crystalline or amorphous (shapeless) solids. But the Ô¬Ārst category was reserved
for large crystals such as gemstones. It was not realized until the present century
that most inanimate materials (and quite a few biological ones as well) consist of
Ô¬Āne crystalline grains, invisible to the eye and not easy to recognize even under a
microscope. Hence they were wrongly classiÔ¬Āed as amorphous. Despite impor-
tant clues, such as the fracture surface of typical metals, crystallinity was regarded
as the exception rather than the rule 11 .
It remains difÔ¬Ācult to distinguish an aggregate of very Ô¬Āne crystals from an
amorphous solid using x-ray diffraction, and careers have been founded on an
increasingly meaningless distinction. Today such amorphous materials as window
glass are accepted as having a random structure, just as John Tyndall suggested
with typical lyricism in Heat‚Ä”A Mode of Motion (1863):

To many persons here present a block of ice may seem of no more in-
terest and beauty than a block of glass; but in reality it bears the same
relation to glass that orchestral harmony does to the cries of the mar-
ketplace. The ice is music, the glass is noise; the ice is order, the glass
is confusion. In the glass, molecular forces constitute an inextricably
entangled skein; in ice they are woven to a symmetric texture (. . . )

Amorphous metals, usually obtained by very rapid cooling from the liquid
state, also have a random structure, in this case approximated by Bernal‚Ä™s random
¬Ĺ¬ľ
See, for instance, Steinhardt P, Jeong H-C, Saitoh K, Tanaka M, Abe E and Tsai A P 1998 Ex-
perimental veriÔ¬Ācation of the quasi-unit-cell model of quasicrystal structure Nature 396 55‚Ä“7; 1999
Nature 399 84.
¬Ĺ¬Ĺ
See for general reference: Lines M E 1994 On the Shoulders of Giants (Bristol: Institute of Physics
Publishing).
The architecture of the world of atoms
90

sphere packings (chapter 3). This structure gives them exceptional properties,
useful in magnetic devices or as a coating on razor blades and in the manufacture
of state-of-the-art golf clubs. What was once a mere academic curiosity now
caresses the chin of the aspiring executive and adds several yards to his drive into
the bushes from the Ô¬Ārst tee.

8.10 Crystal nonsense
Old errors cast long shadows in our conception of the natural world. Astrology
still commands copious column-inches in the daily papers, spiritualists ply a busy
trade, and books abound on the mystical power of crystals. A suitable crystal,
we are told, can radiate its energy to us and inÔ¬‚uence our aura, in a harmonious
vibration of happiness. If only it were so easy. . . . Solid state physicists would go
around with permanent smiles on their faces.
This strange attribution of a hidden potency is derived from the time when
crystals were regarded as rare exceptions to the general disorder of inanimate
nature. Their strange perfection of form must have led primitive man to wonder
at them: it is known that Peking Man collected rock crystals. Perhaps in modern
times the renewed fascination with ‚Ä˜crystal energy‚Ä™ also derives from the period
of early radio when ‚Ä˜crystal sets‚Ä™, consisting of little besides a point contact to a
crystal, acting as a rectiÔ¬Āer, could be used to listen to radio broadcasts. Magic
indeed!
The more sophisticated may have been aware that the details of the process
of growth of a crystal proved intractable to any convincing explanation for some
time. The problem was, for example, mentioned by A E H Tutton (1924) The
Natural History of Crystals:
One of the most deeply interesting aspects of a crystal (. . . ) concerns
the mysterious process of its growth from a solution (. . . ). The story of
the elucidation, as far as it has yet been accomplished, of the nature of
crystallization from solution in water is one of the most romantic which
the whole history of science can furnish.
It is, by its very nature, a delicate problem of surface science, where minute
amounts of impurities, or defects, can do strange things to help or hinder growth.
But, by and large, crystallization is no longer such a mystery‚Ä”otherwise the semi-
conductor industry could hardly be engaged in making huge silicon crystals of
extraordinary perfection, every day.
The progress of physics did not, however, deÔ¬‚ect Rupert Sheldrake from us-
ing the assumed intractability of the explanation of crystallization as a pretext for
his provocative theory of ‚Ä˜morphic resonance‚Ä™, according to which crystallization
is guided by a memory from the past. As a botanist of impeccable pedigree, he
was then able to generalize his principle to the living world, creating a consider-
able following and almost unbearable irritation in the orthodox scientiÔ¬Āc commu-
nity.
Chapter 9

Apollonius and concrete

9.1 Mixing concrete
Any builder knows that to obtain compact packings in granular mixtures such as
the ‚Ä˜aggregate‚Ä™ used to make concrete, the size of the particles must vary over
a wide range. The reason is evident: small particles Ô¬Āt into the interstices of
larger ones, leaving smaller interstices to be Ô¬Ālled, and so on. A typical recipe for
very dense mixtures starts with grains of a given size and mixes them with grains
of smaller and smaller sizes in prescribed ratios of size and quantity, as already
mentioned in chapter 2. The resulting mixture has a density that can approach
unity.
Such recursive packing was already imagined around the 200 BC by Apol-
lonius of Perga (269‚Ä“190 BC), a mathematician of the Alexandrine school. He
is classiÔ¬Āed with Euclid and Archimedes among the great mathematicians of the
Greek era. His principal legacy is the theory of those curves known as conic sec-
tions (ellipse, parabola, hyperbola). He brought it to such perfection that 1800
years passed before Descartes recast it in terms of his new methods.
The method of recursive packing reappeared more recently in a letter by
G W Leibniz (1646‚Ä“1716) to Brosses:

Imagine a circle; inscribe within it three other circles congruent to each
other and of maximum radius; proceed similarly within each of these
circles and within each interval between them, and imagine that the
process continues to inÔ¬Ānity. (See Ô¬Āgure 9.1.)

Something similar also arose in the work of the Polish mathematician, Wa-
claw Sierpi¬ī ski (1887‚Ä“1969), who wrote a paper in 1915 on what has come to
n
be called the ‚Ä˜Sierpi¬ī ski Gasket‚Ä™, and well known as a good example of a fractal
n

91
Apollonius and concrete
92

Figure 9.1. Apollonian packing.

Figure 9.2. Sierpi¬ī ski Gasket.
n

structure (see Ô¬Āgure 9.2). Ian Stewart has called it the ‚Ä˜incarnation of recursive
geometry‚Ä™.
The aim of Sierpi¬ī ski was to provide an example of a curve that crosses itself
n
at every point, ‚Ä˜a curve simultaneously Cartesian and Jordanian of which every
point is a point of ramiÔ¬Ācation‚Ä™. Clearly this curve is a fractal, but this word was
Apollonian packing 93

coined by BenoňÜt Mandelbrot 1 only in 1975.
ń±
Sierpi¬ī ski was exceptionally proliÔ¬Āc‚Ä”he published 720 papers and more
n
than 60 books. He called himself an ‚Ä˜Explorer of the InÔ¬Ānite‚Ä™.

9.2 Apollonian packing
In the packing procedure known as ‚Ä˜Apollonian Packing‚Ä™, one starts with three
mutually touching circles and puts in the hole between them a fourth circle which
touches all three. Then the same procedure is iterated.
Apollonius studied the problem of Ô¬Ānding the circle that is tangent to three
given objects (each of which may be a point, line or circle). Euclid had already
solved the two easiest cases in his Elements, and the other (apart from the three-
circle problem) appeared in the Tangencies of Apollonius. The three-circle prob-
lem (or the kissing-circle problem) was Ô¬Ānally solved by Vi` te (1540‚Ä“1603) and
e
the solutions are called Apollonian circles. A formula for Ô¬Ānding the radius (√– )
of the fourth circle which touches three mutually tangent circles of radii (√– ¬Ĺ , √–¬ĺ
and √–¬Ņ ) was given by Ren¬ī Descartes in a letter in November 1643 to Princess
e
Elisabeth of Bohemia:

¬Ĺ ¬· √–¬Ĺ ¬· √–¬Ĺ ¬· √–¬Ĺ
¬ĺ ¬ĺ ¬ĺ ¬ĺ
¬ĺ √–¬Ĺ ¬ĺ ¬Ņ

¬Ĺ¬·¬Ĺ¬·¬Ĺ¬·¬Ĺ ¬ĺ
(9.1)
√– √– √– √–
¬Ĺ ¬ĺ ¬Ņ

This formula was rediscovered in 1936 by the physicist Sir Frederick Soddy
who expressed it in the form of a poem, ‚Ä˜The Kiss Precise‚Ä™ 2 :

Four pairs of lips to kiss maybe
Involves no trigonometry.
‚Ä™Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.
Four circles to the kissing come,
The smaller are the benter,
The bend is just the inverse of
The distance from the centre.
¬Ĺ
Mandelbrot B B 1977 The Fractal Geometry of Nature (New York: Freeman).
¬ĺ
Soddy F 1936 The kiss precise Nature 137 1021.
Apollonius and concrete
94

Though their intrigue left Euclid dumb.
There‚Ä™s now no need for the rule of thumb.
Since zero bends a straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.
To spy out spherical affairs
An oscular surveyor
Might Ô¬Ānd the task laborious,
The sphere is much the gayer,
And now besides the pair of pairs
A Ô¬Āfth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
The square of the sum of all Ô¬Āve bends
Is thrice the sum of their squares

In the Apollonian procedure, the size of the circles inserted inside the holes
become smaller and smaller and the packing fraction approaches unity in the inÔ¬Ā-
nite limit. For example, one can start from three equal tangent unit circles which
pack with a density of 0.907. . . . Inside the hole one can insert a circle with ra-
dius ¬Ĺ ¬ľ ¬Ĺ and the density becomes 0.95. . . . Now there are three
holes where one can insert three circles with radius ¬Ĺ ¬Ĺ ¬ľ ¬ľ ¬Ņ and
the density rises to ¬ľ .

9.3 Packing fraction and fractal dimension
Pursued indeÔ¬Ānitely, Apollonian packing leads to a dense system with packing
¬Ĺ. But how is this limit reached? Suppose for instance that we
fraction
start with circles of radii √–√ź √– and stop the sequence when the radii arrive at the
minimum value √– √—√‘ √ź√ź . The packing fraction is
¬ī¬ĺ¬† ¬µ
√–√—√‘
¬Ĺ¬† √ź√ź
(9.2)
√–√ź √–

where is the fractal dimension. Indeed the Apollonian packing is a clas-
sical example of a fractal, in which the structure is composed of many simi-
lar components with sizes that scale over an inÔ¬Ānite range. Numerical simu-
¬Ĺ ¬Ņ¬ľ 3 . The analytical determination of is a surpris-
lations give
ingly difÔ¬Ācult problem. Exact bounds have been calculated by Boyd who found
¬Ĺ ¬Ņ¬ľ¬ľ ¬Ĺ ¬Ĺ ¬Ņ¬Ĺ ¬Ņ 4. The fractal dimension can be calculated in the two
¬Ņ
Manna S S and Hermann H J 1991 Precise determination of the fractal dimension of Apollonian
packing and space-Ô¬Ālling bearings J. Phys. A: Math. Gen. 24 L481‚Ä“90.
Boyd D W 1973 Mathematica 20 170.
Packing fraction in granular aggregate 95

Figure 9.3. ‚Ä˜So we may image similar rings of spheres above and below (. . . ) and then
being all over again to Ô¬Āll up the remaining spaces and so on ad inÔ¬Ānitum, every sphere
added increasing the number that have to be added to Ô¬Āll it up!‚Ä™ [Soddy F 1937 The bowl
of integers and the hexlet Nature 139 77‚Ä“9.]

¬Ĺ
interesting models for packings with triangles and hexagons which have
¬Ĺ respectively5.
and

9.4 Packing fraction in granular aggregate
The Apollonian packing procedure can be extended to three dimensions. In this
case, four spheres are closely packed touching each other and a Ô¬Āfth one is in-
serted in the hole between them. Then the procedure continues inÔ¬Ānitely as in
two dimensions.
Descartes‚Ä™ theorem (equation (9.1)) was extended to three dimensions by
Soddy in the third verse of his poem and to dimensions by Gosset in another
poem also entitled ‚Ä˜The Kiss Precise‚Ä™ 6 .

Each with √’ ¬· ¬Ĺ fold mate
The square of the sum of all the bends
Is √’ times the sum of their squares.

Gosset‚Ä™s equation replaces the factor 2 in front of equation (9.1) with a factor
(the space dimension, √’ in Gosset‚Ä™s poem). In three dimensions it gives, for
example, a radius of 0.2247. . . for the maximum sphere inside the hole between
four touching spheres.
The relation for the packing density can also be extended to three and higher
dimensions7 .
¬†
¬ī ¬µ
√–√—√‘
¬Ĺ¬† √ź√ź
(9.3)
√–√ź √–
This expression is not limited to the Apollonian case but is valid for any fractal
√–√—√‘ √ź√ź ) with power law
packing in the limit of very wide polydispersity (√– √ź √–
in the size distribution.
The fractal dimension for Apollonian sphere packings has been less studied
than in the two-dimensional case. However, the two models of packings with
hexagons and triangles can be easily extended to three dimensions giving
¬Ĺ ¬ĺ and ¬ĺ.
Engineers have long known 8 that the porosity √” of the grain mixture is a
function of the ratio between radii of the smallest and the largest grains utilized.
They empirically use the equation
¬Ĺ
√–√—√‘
¬Ĺ¬† √ź√ź
√” (9.4)
√–√ź √–

The Giant‚Ä™s Causeway is a columnar basalt formation on the north coast of Ire-
land. It has been an object of admiration for many centuries and the subject of
continual scientiÔ¬Āc debate 1 . Although it still draws tourists from afar, Dr John-
son‚Ä™s acid remark that it was ‚Ä˜worth seeing but not worth going to see‚Ä™, echoed
by the irony of Thackeray‚Ä™s account of a visit, may be justiÔ¬Āed, since similar ge-
ological features occur throughout the world. Among the more notable examples
are those of the Auvergne in France, Staffa in Scotland and the Devil‚Ä™s Postpile in
the Sierra Nevada of California.
The primary historical importance of the debate on the origins of the Cause-
way lies in it being a focus of the intellectual battle between the Neptunists and
the Vulcanists in the 18th century. The history of geology delights in giving such
titles to its warring sects: others have been classed as Plutonists, Catastrophists
and Uniformitarians.
To the convinced Neptunist the origin of rocks lay in the sedimentary pro-
cesses of the sea, while a Vulcanist would argue for volcanic action. As in most
good arguments, both sides were right in certain cases. But in the case of basalt
the Neptunists were seriously wrong.
Why should this concern us here? Simply because the story is intertwined
with many of the strands of ideas on packing and crystallization in the understand-
ing of materials which emerged over the same period.
What was so fascinating about the Causeway?

Figure 10.1. Sketch of the Giant‚Ä™s Causeway (from Philosophical Transactions of the
Royal Society 1694).

10.2 Idealization oversteps again
As with the bee‚Ä™s cell, but even more so, sedentary commentators on the Cause-
way have generally overstated the perfection of order which is to be seen in the
densely packed basalt columns. They have been described as ‚Ä˜hexagonal‚Ä™, im-
plying that the pattern is a perfect honeycomb. This is not at all the case (see
Ô¬Āgure 10.2).
Eyewitness reports, especially by unscientiÔ¬Āc visitors, were generally more
accurate. The Percy family of Boston reported in their Visit to Ireland (1859):
Do they not all look alike?
Yes, just as the leaves are alike in general construction, but endlessly
diverse, just as all human faces are alike, but all of them possessed of
an individual identity.
But as the story was passed around, idealization constantly reasserted a reg-
The Ô¬Ārst ofÔ¬Ācial report 99

Figure 10.2. Distribution of the number of sides of basalt columns in two of the most
famous sites where these occur (after Spry A 1962 The origin of columnar jointing, partic-
ularly in basalt Ô¬‚ans J. Geol. Soc. Australia 8 191).

ular hexagonal pattern for the Causeway, instead of the elegant random pattern
in which less than half of the polygons are six-sided, as one can verify from Ô¬Āg-
ure 10.2.
‚Ä˜Hexagonal‚Ä™ was an evocative word, calling to mind the form of many crys-
tals. It was natural therefore to call the Causeway ‚Ä˜crystalline‚Ä™; and see the
columns as huge crystals, even though their surfaces were rough in appearance.
Alternatively, it was suggested that the columns are formed by compaction or by
cracking. The latter was the choice of the Vulcanists, who saw the basalt as slowly
cooling and contracting, until it cracked.
That random cracking should have this effect seems almost as unlikely as
crystallization, at Ô¬Ārst glance, yet it has come to be accepted, as we explain later.
This has not stopped the continued generation of wild theories, even late in the
20th century.

10.3 The Ô¬Ārst ofÔ¬Ācial report
In 1693 Sir Richard Bulkeley made a report of the Causeway to the Royal So-
ciety in London. Like many who were to follow, he offered an account of the
phenomenon without troubling himself to go to see it. He relayed the news from
a scholar and traveller well known to him, that it ‚Ä˜consists all of pillars of perpen-
dicular cylinders, Hexagones and Pentagones, about 18 to 20 inches in diameter‚Ä™.
While offering no promise to make a visit himself, he offered to answer
any queries. In the following year Samuel Foley published answers to questions
forwarded by Bulkeley. Already the similarity to crystalline forms was noted. A
more scholarly and verbose account by Thomas Molyneux of Dublin followed,
The Giant‚Ä™s Causeway
100

complete with classical references, and containing some intemperate criticisms of
the original reports. Despite his superior tone, the author confesses that ‚Ä˜I have
never as yet been upon the place myself‚Ä™. He noted a similarity to certain fossils
described by Lister but found the difference of scale difÔ¬Ācult to explain away.
For a time the arguments lapsed, but a number of Ô¬Āne engravings with de-
tailed notes were published. Art served science well, in providing an inspiring
and accurate picture for the armchair theorists of geology. The correspondence
resumed around 1750. Richard Pockock had spent a week at the Causeway, and
settled on a Neptunist mechanism of precipitation.
In 1771, N Desmarest published a memoir which was to be central to the
Vulcanist/Neptunist dispute. In his view the ‚Ä˜regular forms of basalt are the re-
sult of the uniform contraction undergone by the fused material as it cooled and
congealed‚Ä™. This was countered by James Keir, who reasserted the crystalline
hypothesis, drawing on observations of the recrystallization of glass. Admittedly
there was a great difference of scale, but ‚Ä˜no more than is proportionate to the
difference observed between the little works of art and the magniÔ¬Ācent operations
of nature‚Ä™. Here ‚Ä˜art‚Ä™ means craft or industry, for recrystallization was a matter of
intense commercial interest in the attempt to reproduce oriental porcelain.
The Reverend William Hamilton added further support to crystallization
with published letters and a dreadful 100-page poem (‚Ä˜Come lonely Genius of
my natal shore. . . ‚Ä™), published in 1811. This and vitriolic rebuttals of Desmarest
by Kirwan and Richardson did not succeed in reversing the advance of the Vul-
canist hypothesis. It was, said Richardson, an ‚Ä˜anti-Christian and anti-monarchist
conspiracy‚Ä™, since it set out to ‚Ä˜impeach the chronology of Moses‚Ä™. He favoured
a model of compression of spheroidal masses to form columns, with some labo-
ratory experiments to back it up.
It was probably fair comment when Robert Mallett summarized the state of
play in 1875 by saying that ‚Ä˜no consistent or even clearly intelligible theory of the
production of columnar structure can be found‚Ä™.
Mallett was an early geophysicist, who invented the term ‚Ä˜seismology‚Ä™. The
name of his engineering works still adorns the railings of Trinity College Dublin.
He undertook a thorough review of the basalt question and attempted to publish
it in the Proceedings of the Royal Society. After Ô¬Āve months and four referees he
was told that ‚Ä˜it was not deemed expedient to print it at present‚Ä™‚Ä”a splendidly
diplomatic refusal to publish the work of a Fellow of the Society.
This reversal may well have stemmed from his trenchant criticism of ‚Ä˜very
crude and ill-thought-out notions‚Ä™ and a ‚Ä˜bad or imperfect experiment inaccurately
reasoned upon and falsely applied‚Ä™ by his predecessors, who were ‚Ä˜blinded by a
preconceived and falsely based hypothesis‚Ä™. His own advocacy was directed in
support of contraction and cracking. He gave credit for this to James Thomson
(the Glasgow professor who was the father of Lord Kelvin) together with the
French school of Desmarest.
He Ô¬Ānally succeeded in publishing his article in the lesser (and less conser-
vative) journal Philosophical Magazine.
Mallett‚Ä™s model 101

Figure 10.3. Polyhedral basaltic columns in the Giant‚Ä™s Causeway.

10.4 Mallett‚Ä™s model

Mallett proposed to attack the problem ‚Ä˜in a somewhat more determinate manner‚Ä™.
By this he meant that his approach would be mathematical and quantitative, in
contrast to the hand-waving of the other geologists. This more modern style has
made the article inÔ¬‚uential ever since.
One of his principal interests was in the energy liberated in volcanic erup-
tions, so it was natural for him to think in terms of the total energy of the system
of cracks rather than their precise mechanism of formation. At a time when there
was a general tendency to express the laws of physics as minimal principles, he
appealed rather vaguely to the ‚Ä˜principle of least action‚Ä™ and ‚Ä˜the minimum ex-
penditure of work‚Ä™. What crack pattern would minimize energy?
This question makes little sense unless some constraint Ô¬Āxes the size of
the cells of the pattern, but this was somehow ignored, and he triumphantly an-
nounced that the hexagonal pattern was best, by comparison with other simple
cases.
The Giant‚Ä™s Causeway
102

10.5 A modern view
D‚Ä™Arcy Wentworth Thompson recognized that cracks which proceed explosively
from isolated centres could never form such a harmonious pattern. He failed to
see the possibility that crack patterns Ô¬Ārst formed deep within a lava Ô¬‚ow could
propagate very slowly outwards as it cooled. A careful reading of Mallett‚Ä™s pa-
per shows that he had already recognized this, and this part of his description is
impeccable.
In their slow motion the cracks migrate until they form a balanced network
which propagates unchanged. To have this property it need not be ordered: the
best analogy is the arrangement of atoms when a liquid becomes a glass on cool-
ing. There is local order only.
When precisely the realization dawned more generally that it must be so is
not clear, but certainly Cyril Stanley Smith gave this explanation in 1981, when
preparing the published version of a lecture to geologists. It appears very natural
to the modern mind. Until full computer simulations are performed, one hesitates
to state it emphatically and risk joining such a long list of unwarranted claims.

10.6 Lost city?
In October 1998, an article in the British press reported the possible discovery of
a lost city by a documentary Ô¬Ālm-maker in Nicaragua. The evidence consisted
of 62 polygonal basalt columns. Troops had been dispatched to guard the Ô¬Ānd
against looters. Geologists had expressed some scepticism. . . .
It seems that the columnar basalt story, begun in 1693, is destined to run and
run.
Chapter 11

Soccer balls, golf balls and
buckyballs

11.1 Soccer balls
A favourite close-up of the television sports director shows a soccer ball distend-
ing the net. This offers the opportunity to compare the two: for the net is nowa-
days made in the form of the hexagonal honeycomb, and the surface of the ball
looks roughly similar. Closer examination reveals the presence of 12 pentagons
among the hexagons on the ball.
The problem of the soccer ball designer was to produce a convenient poly-
hedral form which is a good approximation to a sphere. The presently favoured
design replaces a traditional one and we are not aware of the precise arguments
that brought this about. Certainly it is more aesthetic, on account of its high sym-
metry, which is ofÔ¬Ācially described as icosahedral. The simplest design of this
type would be that of the pentagonal dodecahedron (Ô¬Āgure 5.6), but this was per-
haps not sufÔ¬Āciently close to a sphere. Instead, 32 faces are used, of which 12 are
pentagons and the rest are hexagons.
By a curious coincidence, this icon of modern sport has cropped up in a
prominent role in modern science as well, as we shall see shortly. But Ô¬Ārst let us
switch sports and examine the golf ball.

11.2 Golf balls
The dynamics of a sphere immersed in a viscous Ô¬‚uid presents one of the classic
set-piece problems of physics and engineering, dating back to the work of Newton
(applied, in particular, to the motion of the Earth through the ether), and tidied

103
Soccer balls, golf balls and buckyballs
104

Figure 11.1. A soccer ball.

Figure 11.2. A golf ball.

up in some respects by Sir George Gabriel Stokes more than 100 years ago. In
modern times it can be safely assumed that there has been a large investment in
a better understanding of the motion of a sphere in air, since it commands the
attention of many important people on the golf courses of the world.
The extraordinary control of the golf ball‚Ä™s Ô¬‚ight which is exercised by
(some) golfers owes much to the special effects imparted by the spin which is
imposed on the ball by the angled blade of the club. This can be as much as
10 000 revolutions per minute. Because of its backspin the ball is subject to an
upward force, associated with the deÔ¬‚ection of its turbulent wake. This is the
Magnus Effect, also responsible for the swerve of a soccer ball. It causes the ball
to continue to rise steeply until, both velocity and spin having diminished, it drops
almost vertically onto its target.
It was found that marking the surface of the ball enhanced the effect, and
a dimple pattern evolved over many years. Today such patterns typically consist
of 300‚Ä“500 dimples. All are consistent with the incorporation of a ‚Ä˜parting line‚Ä™
where two hemispherical moulds meet to impress the shape. Locally the dimples
are usually close-packed on the surface of the ball.
Buckyballs 105

We know of no physical theory which would justify any particular arrange-
ment. Many are used, whether motivated by whim or the respect for the intellec-
tual property of established designs. Titleist has favoured an essentially icosahe-
dral ball derived from the pentagonal dodecahedron by adding hexagons, just as
for the soccer ball. Note that here we are dealing with a ‚Ä˜dual‚Ä™ structure: each
dimple lies at the centre of one of the polygons. A simple topological rule gov-
erns all such patterns. Indeed, to wrap a honeycomb on a sphere or other closed
surface is not possible, without introducing other kinds of rings. According to
Euler‚Ä™s theorem the minimum price to be paid is 12 pentagons. One may take the
pentagonal dodecahedron, beloved of the Greeks, and expand it by the addition
of any number of hexagons (except, as it happens, one). In some cases the result
is an elegantly symmetric structure.

11.3 Buckyballs
The modern science of materials has matured to the point at which progress seems
barred in many directions. One cannot envisage, for example, magnetic solids
which are much more powerful than the best of today‚Ä™s products, because they
are close to very basic theoretical limits. (The scientist who says this type of
thing is always in danger of following in the footsteps of the very great men who
denounced the aeroplane, the space ship and the exploitation of nuclear energy as
patent impossibilities.)
Despite this sense of convergence to a state in which optimization rather than
discovery is the goal, new materials continue to make dramatic entrances. Some-
times they arise because certain assortments of many elements have not previ-
ously been tried in chemical combination. There is one great exception to this
trend towards combinatorial research. The sensational advance in carbon chem-
istry which goes by the affectionate nickname of ‚Ä˜buckyballs‚Ä™ has raised more
eyebrows and opened more doors than anything else of late, with the possible
exception of high-temperature superconductors.
This is not a case of a single, momentous revelation: rather one of steadily
increasing knowledge and decreasing incredulity over many years, from the Ô¬Ārst
tentative clues to the establishment of major research programmes throughout the
world. The story up to 1994 has been compellingly recounted by Jim Baggott in
Perfect Symmetry1 .
At its conclusion Baggott wondered which of the four central personalities
of his tale would be rewarded by the Nobel Prize, which is limited to a trio. The
answer came in 1996: the Royal Swedish Academy of Science awarded the Nobel
Prize in Chemistry to Robert F Curl, Harold W Kroto and Richard E Smalley.
The carbon atom has long been renowned as the most versatile performer in
the periodic table. It is willing to join forces with other atoms either three or four
¬Ĺ
Baggott J 1994 Perfect Symmetry: The Accidental Discovery of Buckminsterfullerene (Oxford: Ox-
ford University Press).
Soccer balls, golf balls and buckyballs
106

Figure 11.3. The C buckyball.
¬ľ

at a time. Pure carbon with fourfold bonding is diamond, whereas graphite con-
sists of sheets with the honeycomb structure in which there is threefold bonding.
The two solids have vastly different properties; one is hard, the other soft; one
is transparent, the other opaque; one is horribly expensive, the other very cheap.
Nothing could better illustrate the falsity of the ancient idea that all the properties
of elements spring directly from the individual atoms.
For pure carbon, that was supposed to be the end of the story, give or take
a few other forms to be found under extremely high pressures. Yet we now rec-
ognize that graphite-like sheets can be wrapped to form a spherical molecule of
60 atoms‚Ä”the buckyball‚Ä”and buckyballs can be assembled to form an entirely
new type of carbon crystal, with startling properties. And further possibilities
continue to emerge in the laboratory or the fevered imagination of molecule de-
signers: other, larger molecules, concentric molecules like onions, tubular forms
called nanotubes, which may be key components of future nanoengineering. The
buckyball belongs to an inÔ¬Ānite family of possibilities which comprise the new
subject of fullerene chemistry.
The buckyball has a great future, a fascinating history and even an intriguing
prehistory. The existence of molecules like this had been teasingly conjectured
by a columnist in the New Scientist, and Buckminster Fuller built most of his rep-
utation on the architectural applications of such structures (generally containing
many more hexagons than the buckyball). Hence the C ¬ľ molecule was Ô¬Ārst bap-
tised buckminsterfullerene in his honour. This is by no means a large mouthful by
Buckminster Fuller 107

chemical standards, but the snappier ‚Ä˜buckyball‚Ä™ has steadily gained currency at
its expense2 .

11.4 Buckminster Fuller
Buckminster Fuller3 has been described by an admirer as a ‚Ä˜protean maverick‚Ä™.
For many he is the prime source of insight into many of the structures which we
have pondered in this book. Over several decades he poured forth a torrent of
ideas and assertions which combined the Greek faith in geometry as lying at the
heart of all nature with vague and superÔ¬Ācially impressive notions of energy and
synergy. The resulting pot-pourri is inspiring or bewildering, according to taste.
When he ventures into fundamental descriptions of nature, it is reminiscent of the
speculations of those 19th-century ether theorists.
One of Fuller‚Ä™s assertions was that all nature is ‚Ä˜tetrahedronally coordinated‚Ä™.
Here he was thinking of close-packing of spheres (although this, in part, con-
tradicts the statement‚Ä”not all arrangements are tetrahedral). He seems to have
claimed to have discovered the ideal close-packed arrangement, only to Ô¬Ānd it
in the work of Sir William Bragg, who he then supposed to have independently
found it around 1924!
It is the practical outcome of Fuller‚Ä™s ruminations, in the form of the geodesic
dome, that we remember today. Here again there is some question of priority.
Tony Rothman, in Science a la Mode, has pointed out that such structures were
`
patented by the Carl Zeiss company, for the construction of planetariums, in the
1920s. Fuller‚Ä™s patent is dated 1954. Rothman generously gives the protean mav-
erick the beneÔ¬Āt of the doubt. . . .

11.5 The Thomson problem
Another problem which involved placing points on a sphere was posed by J J
Thomson in 1904 in the context of speculations about classical models of the
atom, which were soon to be rendered out-of-date by quantum mechanics. But as
with many other cases in this book, the problem has survived in its abstract math-
ematical form, and continues to intrigue mathematicians and challenge computer
scientists. It is simply this: what is the arrangement of √Ü point electrical charges
on a sphere, which minimizes the energy associated with their interactions? This
is just the sum of ¬Ĺ √– over all pairs of points, where √– is their separation. Gen-
erally speaking the solution is neither the best packing nor the most symmetric
arrangement of points. Beginning with L Foppl, around 1910, mathematicians
¬ĺ
It has been suggested that, since the polyhedral structure of the buckyball and soccer ball date back
to Archimides, ‚Ä˜archiball‚Ä™ might be more appropriate.
¬Ņ
See, for reference, Marks R W 1960 The Dymaxion World of Buckminster Fuller (Reinhold); Roth-
man T 1989 Science a la Mode, Physical Fashions and Fictions (Princeton, NJ: Princeton University
`
Press).
Soccer balls, golf balls and buckyballs
108

have accepted Thomson‚Ä™s challenge. Kusner and Sullivan 4 analysed the cases
¬ĺ¬ľ, discovering that there is only one stable structure when √Ü is smaller
√Ü
than 16.
The target of most researchers is to Ô¬Ānd structures of low energy using com-
putational search procedures and identify the one which is lowest among these,
for large values of √Ü 5 . Some of the structures which crop up here are similar to
those of Buckminster Fuller constructions and golf balls.
This computational quest is an ideal testing-ground for new software ideas,
such as simulated annealing (see section 13.11).
The minimal energy structures for small √Ü are surprising in some cases: for
√Ü we do not Ô¬Ānd the obvious arrangement in which the charges are at the
corners of a cube, but rather a twisted version of this.
¬Ĺ¬ĺ the familiar icosahedral structure is found, with the charges at
For √Ü
the corners of the pentagonal dodecahedron (Ô¬Āgure 5.6). Each charge has Ô¬Āve
nearest neighbours. Thereafter this structure is elaborated to accommodate more
charges. For all √Ü which satisÔ¬Āes
¬Ĺ¬ľ¬ī√‘¬ĺ ¬· √’¬ĺ ¬· √‘√’¬µ ¬· ¬ĺ
√Ü (11.1)
with √‘ and √’ being positive integers; this can be done very neatly as in Ô¬Āg-
ure 11.4.
All the additional charges have six neighbours. In special cases these corre-
spond to the buckyball or soccer ball structure which we have already admired.
That is not the end of the story. Eventually at high √Ü these structures can
be improved by modiÔ¬Ācations which introduce more charges with Ô¬Āve or seven
neighbours.

11.6 The Tammes problem
Many pollen grains are spheroidal and have exit points distributed on the surface.
The pollen comes out from these points during fertilization. The position of the
exit points is rather regular and the number of them varies from species to species.
In 1930 the biologist Tammes described the number and the arrangement of the
exit points in pollen grains of many species. He found that the preferred numbers
are 4, 6, 8, 12, while 5 never appears. The numbers 7, 9 and 10 are quite rare and
11 is almost never found. He also found that the distance between the exit points
is approximately constant, and the number of these points is proportional to the
surface of the sphere 6 .
Tammes posed the following question: given a minimal distance between
them, how many points can be put on the sphere? We can think of the points as
Kusner R and Sullivan J 1997 Geometric Topology ed W H Kazez (International Press).
Altschuler E L et al 1997 Possible global minimum lattice conÔ¬Āgurations for Thomson‚Ä™s problem
of charges on a sphere Phys. Rev. Lett. 78 2681.
Tammes P M L 1930 On the origin of number and arrangement of the places of exit on pollen grains
Diss. Groningen.
The Tammes problem 109

Figure 11.4. Three low-energy structures for charges on a sphere as depicted by Altschuler
et al.

associated with (curved) discs of a certain size, which are not allowed to overlap.
Tammes attacked the problem in an empirical way by taking a rubber sphere and
drawing circles on it with a compass. He found, for instance, that when the space
is enough for Ô¬Āve circles then an extra circle can always be inserted. In this
case the six circles are located at the vertices of an octahedron. In this way the
preference for 4 and 6 and the aversion for 5 in pollen grains may be explained.
Tammes also found that when 11 points Ô¬Ānd enough space then 12 can also be
placed at the vertices of an icosahedron.
The Ô¬Ārst of these results has been mathematically proved to be valid for the
surface of a sphere in three dimensions and it has been extended to any dimension.
If on the surface of a sphere in -dimensional space more than ¬· ¬Ĺ discs can be
placed then ¬ĺ such discs can be placed at the extremities of the coordinate axes.
The Tammes problem is the subject of an enormous amount of literature.
Soccer balls, golf balls and buckyballs
110

Mathematically the questions raised by Tammes can be expressed as follows:
what is the largest diameter √Ü of √Ü equal circles that can be placed on the
surface of a unit sphere without overlap? How must the circles be arranged, and
is there a unique arrangement?
¬Ĺ¬ĺ and √Ü ¬ĺ . These and other
Exact solutions are known only for √Ü
solutions are shown in table 11.1 (taken from Croft 7 ).

√Ü ¬†¬Ĺ which is the Tammes empirical
Note that for √Ü and 12, √Ü
result.
A bound for the minimum distance between any pair of points on the sur-
face of the unit sphere, was given in 1943 by Fejes T¬ī th
o
√—
√Ü
¬† cosec ¬ĺ
¬ī√Ü ¬† ¬ĺ¬µ (11.2)

¬Ņ
with the limit exact for √Ü and 12 8 .

11.7 Helical packings
The packing of spheres around a cylinder results in helical patterns, which may of-
ten be seen in street festivals when balloons are used to decorate lamposts. These
attractive structures have an interesting history because they are found in many
plants. Botanists have long been fascinated by the way in which branches or
Croft H T, Falconer K J and Guy R K 1991 Unsolved Problems in Geometry (New York: Springer)
p 108.
Ogilvy C S 1994 Excursions in Mathematics (New York: Dover) p 99.
Helical packings 111

Figure 11.5. Carbon crystals make tubular structures called ‚Ä˜nanotubes‚Ä™. In the tubular
part carbons make a network with rings of six atoms, whereas the positive curvature is
induced by rings of Ô¬Āve atoms.

leaves are disposed along a stem, or petals in Ô¬‚ower. They have found many dif-
ferent helical arrangements, but almost all have a strange mathematical property,
which is the main preoccupation of much of the extensive literature on this sub-
ject, at times acquiring a mystical Ô¬‚avour. This dates back (at least) to Leonardo.
In the 19th century, the Bravais brothers, and later Airy and Tait began a more
modern study, which surprisingly continues today. The reviewer of a recent book
by Roger Jean, said that it ‚Ä˜remains one of the most striking phenomena of biol-
ogy‚Ä™.
Recently the subject has recurred in an exciting new context‚Ä”the creation
of nanotubes, which are tiny tubes with walls which are a single layer of carbon
atoms. They are Ô¬Ārst cousins to the ‚Ä˜buckyballs‚Ä™ (see section 11.3). They were
Ô¬Ārst made in the 1970s by Morinobu Endo, a PhD student at the University of
Orleans.
Biologists call this subject ‚Ä˜phyllotaxis‚Ä™, and obscure it further with terms
such as ‚Ä˜parastichies‚Ä™, which are rather repellent to the Ô¬Ārst-time reader. But
these helical structures are really quite simple things. If we are to place spheres
or points on the surface of a cylinder, it is much the same as placing them on
a plane. We therefore expect to Ô¬Ānd the close-packed triangular arrangement of
chapter 2, which is optimal under a variety of conditions. The difference lies in
the fact that the surface is wrapped around the cylinder and joins with itself. It is
like wallpapering a large pillar‚Ä”the packing must continue smoothly around the
cylinder, without interruption.
We could think of cutting out a strip from the triangular planar packing and
wrapping it around the cylinder. We might have to make some adjustments to
avoid a bad Ô¬Āt where the edges come together. We can displace the two edges
with respect to each other and/or uniformly deform the original pattern (not easy
with wallpaper!), in order to have a good Ô¬Āt.
Let us reverse this train of thought: roll the cylinder across a plane, ‚Ä˜printing‚Ä™
its surface pattern again and again. We expect to get the close-packed pattern, or a
Soccer balls, golf balls and buckyballs
112

Figure 11.6. Bubbles packed in a cylinder show the familiar hexagonal honeycomb
wrapped on a cylinder. This simulation by G Bradley shows surface patterns and the inte-
rior for one of these structures.

strained version of it. The three directions of this pattern, in which the points line
up, correspond to helices on the cylinder. Taking one such direction, how many
helices do you need to complete the pattern? This deÔ¬Ānes an integer, and the three
integers , √ź, √‘ corresponding to the three directions can be used to distinguish
this cylindrical pattern from all others. It is easily seen that two of these must add
to give the third integer.
This is the notation of phyllotaxis, applied to plants, to nanotubes and to
bubble packings within cylinders. In the last case, the surface structure is always
of this close-packed form.
Chapter 12

Packings and kisses in high
dimensions

12.1 Packing in many dimensions
The world of mathematics is not conÔ¬Āned to the three dimensions of the space that
we inhabit. Mathematicians study sphere-packing problems in spaces of arbitrary
dimension. Geometrical puzzles can be posed and solved in such spaces. Some
practical challenges end up in such a form.
With this chapter we present a short excursion into some of the relevant
points concerning packings of spheres in high dimensions and their applications.
The topic is explored in a very comprehensive way by Conway and Sloane in their
book Sphere Packings, Lattices and Groups 1 , which is considered the ‚Ä˜bible‚Ä™ of
this subject.
Packings in many dimensions Ô¬Ānd applications in number theory, numeri-
cal solutions of integrals, string theory, theoretical physics and digital commu-
nications. In particular, some problems in the theory of communications, with
a bearing on the optimal design of codes, can be expressed as the packing of -
dimensional spheres. Indeed, in signal processing it is convenient to divide the
whole information into uniform pieces and associate each piece with a point in
a -dimensional space (a point in a -dimensional space is simply a string of
real numbers √™ ¬Ĺ √™¬ĺ √™¬Ņ √™ ). To transmit and recover the information in
the presence of noise one must ensure that these points are separated by a dis-
tance larger than that at which the additional noise would corrupt the signal. Each
point (encoded information) can be seen as surrounded by a Ô¬Ānite volume, a -
dimensional ball with a diameter larger than the additional noise. The encoded
¬Ĺ
Conway J H and Sloane N J A 1988 Sphere Packings, Lattices and Groups (Berlin: Springer).